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;xpenmenta Chaos Conferenc<
.ouis M. Pecora Mark L. Spano
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World Scientific
Proceedings of the
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;xpenmenta Chaos Conferenc
E2. For the sample with a barrier width of 4.0 nm, the fundamental oscillation frequency is about 10 MHz. For sample 2, which has a thinner barrier of 2.7 nm, the oscillation frequency has increased to 95 MHz. We conclude that the frequency of the oscillations can be varied by changing the barrier width. Note that in the frequency spectra on the right side in Fig. 7 sample 1 contains a number of higher harmonics, while the oscillation in sample 2 is much more sinusoidal. We have investigated a number of samples with different barrier widths. 8 ' 11 The measured frequency fs of the spontaneous oscillations for different samples and for different current plateaus within a single sample are shown in Fig. 8 on a logarithmic scale as a function of the exponent of resonant coupling C in the WKB approximation, i.e., ^2m*(V -E) (4) h where m* denotes the effective mass of the electron, V the height of the barrier, and E the energy of the injecting subband. We clearly observe the expected exponential C =
0.2 0.1
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400
B
< He-flow cryostat (4.2 - 300 K) with 20 GHz coaxial cables
Figure 6. Experimental configuration for measuring spontaneous current oscillations and driven chaotic oscillations.
200 0
200
400
Frequency (MHz) Figure 7. Time traces (left) and frequency spectra (right) of spontaneous current oscillations in sample 1 (top) for —8 V and sample 2 (bottom) for -7.76 V recorded at 5-6 K.
20 dependence of the W K B approximation. The dashed line shows a least square fit to the data points, resulting in a slope of —1.3 so that empirically fs = A exp(—1.3 C). We conclude that we can tune the frequency over a wide range by changing the barrier width. However, even within a single sample, we can vary the frequency over more than an order of magnitude by changing the applied field from the first to a higher plateau, which reduces the effective barrier height. In addition to the frequency of the recycling motion of the domain boundary over many superlattice periods, there is another response present with a much higher frequency. 8 ' 12 - 13 In Fig. 9, a current trace recorded in the first plateau of sample 1 at 6 K is shown, which contains a number of spikes with a frequency of 10 MHz in addition to the fundamental oscillation frequency of 0.65 MHz. These spikes are due to the relocation of the domain boundary over a single period. There are 16 spikes between two maxima of the 0.65 MHz oscillation indicating that the recycling motion of the charge accumulation layer covers about 40% of the total superlattice thickness. By introducing a delay time in the drift velocity in Ampere's law, spikes also appear in the calculated current versus time traces. The calculated field distribution shows that this high-frequency component is due to the relocation of the domain boundary by a single period. Other samples also show this effect. The fraction of the superlattice involved in the recycling motion varies between 30 and 40%. Spontaneous current oscillations have also been observed in undoped photoexcited indirect-gap, 14 i.e., type II, and direct-gap, i.e., type-I, superlattices 15 "" 18 using a laser for photoexcitation to introduce the carrier density. The advantage
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2 d„ V2 m* (V-E) /B Figure 8. Oscillation frequency versus resonant coupling exponent C for a number of GaAs/AlAs superlattices (solid symbols) and one Ino.53Gao.47As/Ino.52Alo.4sAs superlattice (open symbol) and for the indicated current plateaus.
Figure 9. Spontaneous current oscillations in sample 1 for an applied voltage of 2.77 V and a temperature of 6 K. There are 16 spikes in a single period of the 0.65 MHz oscillations.
21 of this system is that the carrier density can be tuned over a wide range within a single sample. Recently, the controllability of the bifurcation by means of the carrier density was experimentally observed. 19 ' 20 5. C h a o t i c Current Oscillations When in addition to a dc voltage a modulating ac voltage is applied to this nonlinear system, different routes to chaos have been calculated 21,22 and experimentally observed. 23 ' 24 We will focus in terms of the dc voltage on the second plateau of the I-V characteristics of sample 1, which is shown in Fig. 10. There are two dc voltages marked in the second plateau, one near the center of the plateau Vd\ and one near the edge Vd2c. Note that, in order to clearly show both plateaus in one plot, the current scale is logarithmic. We then applied an ac driving voltage at a frequency fd of the golden mean [(l + v / 5)/2 = 1.618] times the fundamental frequency fs of the spontaneous current oscillations. For Vd\, fd was set to 18.4 MHz, while for Vdc we used 49.4 MHz. We varied the amplitude of the ac modulation voltage Vac and recorded time traces as well as frequency spectra. In Fig. 11, several time traces are shown at Vd\ for different values of Vac. At Vac = 75 mV, the trace clearly exhibits frequency locking with a winding number of 5/8, while at Vac — 180 mV a 2/3 frequency-locked state has been reached. At Vac = 175 mV, the time trace shows some signature of chaos. In Fig. 12, selected time traces at Vdc for different values of Vac = 34, 40, and 63 mV exhibit a quasi-periodic, 2/3 frequency-locked, and chaotic state, respectively. All these time traces show only a small fraction of the recorded total time scale. We
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Figure 10. I-V characteristics of sample 1 at 5 K. Two voltages are marked V^c = 7.08 and Vjc = 6.574 V, the first one near the center and the second one near the edge of the second plateau.
0.0
0.4
0.8
1.2
1.6
Time (us) Figure 11. Driven current oscillations for an applied voltage of Vjc = 7.08 V in sample 1 recorded at 5 K for several amplitudes of the ac driving voltage as indicated.
22
typically measured up to 1600 periods with a resolution of 20 points per period. Power spectra of the driven oscillations are shown in Figs. 13 and 14 for the dc voltages Vjc and V%., respectively. For Vjc, there are alternating windows of quasi-periodicity and frequency locking. The frequency-locked windows starting at Vac = 75 mV exhibit the following sequence of winding numbers as defined by the ratio of the number of frequencies up to the intrinsic frequency and the total number of frequencies up to the driving frequency, 5/8, 7/11, 9/14, (2n+l)/(3n+2) reaching at very high values of Vac 2/3. This number also reflects the ratio of the driving frequency to the intrinsic frequency at that particular value of Vac- Note that the intrinsic frequency increases slightly with increasing Vac- The quasi-periodic windows are characterized by a much richer frequency spectrum. For quasi-periodicity, the ratio of the driving to the intrinsic frequency equals an irrational number. The larger number of frequencies appears due to linear combinations of the these two frequencies. However, in the range of Vac = 170 to 180 mV, the frequency spectra become rather smeared out so that the oscillations do not correspond to a quasiperiodic or frequency-locked state. The time trace at Vac — 175 mV looks at first glance very similar to the one at Vac = 180 mV. A closer look reveals a clear difference between the two traces. The relative amplitude of the two smaller peaks remains constant for Vac = 180 mV, while it varies continuously for Vac = 175 mV. Furthermore, around 1.2 [is, there is a strong change in the amplitude of the current oscillations (cf. Fig. 11). Both observations indicate that in this range of the driving voltage amplitude chaotic oscillations are present. The frequency power spectra at V^c = 6.574 V shown in Fig. 14 exhibit a very different dependence on Vac- At low values of Vac, a quasi-periodic regime is
quasiperiodic
2/3 locking
S
75
100
125
V (mV) Figure 12. Driven current oscillations for an applied voltage of Vjc = 6.574 V in sample 1 recorded at 5 K for several amplitudes of the ac driving voltage as indicated.
Figure 13. Power spectra of the driven current oscillations as a function of the driving voltage amplitude Vac for Vd\ = 7.08 V. The darker the area in the plot, the larger the absolute value of the amplitude of the oscillations.
23
observed, which ends at 37 mV. At this point, the frequency spectra begin to smear out over a limited spectral range. At Vac = 40 mV, a frequency-locked state with a winding number 2/3 appears, which extends up to 50 mV. For larger values of Vac, the frequency spectra are more or less smeared out over the whole frequency range indicating the presence of chaotic oscillations. At Vac = 100 mV, only a 1/1 frequency-locked state remains. The large difference of Figs. 13 and 14 are probably due to the fact that for a dc voltage near the edge of the plateau and above a certain driving voltage amplitude the transition from the oscillating to the static state strongly influences the response of the system to the ac modulation. For values of Vac above 50 mV, the system may enter and leave the static field distribution periodically resulting in the observed complex behavior. In order to obtain further insight in the chaotic behavior, we have analyzed time traces in terms of a return map, i.e., we plot the value of the current at the n + Ist period as a function of the nth period using the driving frequency as the sampling frequency. These so-called Poincare maps are shown for selected values of Vac in Figs. 15 and 16 for the dc voltages VJC and V}c, respectively. In the case of a dc voltage applied near the center of the plateau, we observe at 75, 125, 145, 155, and 180 mV 8, 11, 14, 17, and 3 points, respectively, as expected for frequency locking with the corresponding winding numbers 5/8, 7/11, 9/14, 11/17, and 2/3. For the quasi-periodic regime between 75 and 125 mV, two examples are shown for Vac — 96 and 111 mV, which consist of a closed loop of points. A closed loop indicates the presence of quasi-periodicity. However, the loop is twisted in comparison to the Poincare map of a single frequency oscillation driven by another frequency with an irrational frequency ratio. This twisting can be explained, when
Vac(mV) Figure 14. Power spectra of the driven current oscillations as a function of the driving voltage amplitude Vac for Vjc = 6.574 V. The darker the area in the plot, the larger the absolute value of the amplitude of the oscillations.
Figure 15. Poincare maps of the driven current oscillations for V\c = 7.08 V and different values of the driving voltage amplitude Vac as indicated.
24
the higher harmonics of the intrinsic oscillations are taken into account. It is also present for the frequency-locked state and can be explained in the same way. At Vac = 170 mV, the closed loop begins to break up. In the range between 170 and 180 mV, we do not observe a closed loop as for quasi-periodicity or isolated points as for frequency locking so that we conclude that chaotic oscillations are present in this regime. The route to chaos follows the well-known path of quasi-periodicity —>• frequency locking —» quasi-periodicity —• ... —• frequency locking —> chaos. As expected from the frequency spectra under external driving voltage, also the Poincare maps for a dc voltage near the edge of the plateau differ significantly from the ones for a dc voltage near the center of the plateau. Up to Vac = 37 mV, typical Poincare maps corresponding to quasi-periodic behavior are observed. Two examples are shown in Fig. 16 for 9 and 34 mV, which have the typical shape. Note that for this dc voltage, the intrinsic oscillations do not contain any higher harmonics. A very different type of Poincare map is derived for 38 and 39 mV. This type of map shows some signature of synchronized chaos. Between 39 and 40 mV, there is a very abrupt change from a spatially extended Poincare map to three isolated points indicating 2/3 frequency locking. This frequency-locked state remains up to Vac = 50 mV. For Vac larger than 50 mV, the Poincare maps are spatially extended with various shapes, which are clearly not related to a quasiperiodic or frequency-locked state. The Poincare maps support the interpretation of the frequency spectra that in this regime of driving voltage amplitudes the driven oscillations are chaotic. When Vac is above 80 mV, the Poincare map consists of an extended spot, which shrinks in size with increasing Vac. At 100 mV, the 1/1 frequency-locked state is reached with a Poincare map consisting of a single point. In this case, the route to chaos follows a different path, quasi-periodicity —>
i
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25
synchronized chaos —*• frequency locking —> chaos —> more complex chaos. In order to obtain information about the dimension of the attractor, we have derived the capacity D 0 , information Di, and correlation dimension D 2 from the recorded time traces.25 Figure 17 shows the dependence of these three dimensions as a function of driving voltage amplitude for V^c. In the quasi-periodic regime for values of Vac below 38 mV, all three dimensions are about 1, which corresponds to the dimension of a closed loop with a constant point density. This value supports the interpretation of this regime as quasi-periodic. However, the values are actually somewhat larger than one. A closer look at the corresponding Poincare maps reveals that there are probably two closed loops on top of each other. This regime may contain a more complicated dynamical behavior than straight quasiperiodicity. Between 40 and 50 mV, all three dimensions are zero indicating the presence of the already identified 2/3 frequency-locked state. For the region of synchronized chaos at 38 to 39 mV, the capacity dimension D0 — 0.64 is significantly smaller than one, but still finite. For Vac larger than 50 mV, all three dimensions strongly increase to values larger than one. A more detailed analysis, in particular of the chaotic regime, is presently performed, which will give further insight in the underlying chaos. Furthermore, a detailed theoretical investigation for dc biases near the center and edge of the plateau is necessary in order to completely understand the chaotic behavior of this system. Finally, we should mention that we also observed undriven chaotic oscillations in doped12'26 and undoped, photoexcited superlattices. 15-18 Presently, the appearance of these undriven chaos is not understood. However, it may be related to the existence of another time scale such as the delay time, which was introduced to explain the current spikes within a single fundamental oscillation. We also studied the response of an undoped, photoexcited superlattice to an external ac driving voltage.16 A very different bifurcation pattern is observed in comparison to the doped superlattices and the calculated results. 6. Summary and Conclusions Weakly coupled superlattices represent a nonlinear system due to sequential resonant tunneling between different subbands in adjacent wells. Several regions of negative differential velocity exist in such systems resulting in the formation of static and dynamic electric-field domains. In the dynamic regime, spontaneous current oscillations appear with frequencies ranging from the sub-MHz regime to several GHz. The frequencies are mainly determined by the resonant coupling between adjacent wells, which depends exponentially on the barrier width as well as the square root of the effective barrier height. When an ac modulation voltage is applied, frequency-locked, quasi-periodic and chaotic oscillations can be observed. Depending on the position of the applied dc bias with respect to the
26 current plateau, different routes to plexity of the route. Finally, in the dimension of the chaotic attractors traces supporting the interpretation
chaos are observed, which differ in the comcase of more complex chaos, the multifractal has been derived from the real-time current derived from the Poincare maps.
7. A c k n o w l e d g m e n t s The author would like to thank H. Asai, L. L. Bonilla, O. Bulashenko, A. Fischer, R. Hey, J. W. Kantelhardt, J. Kastrup, K. J. Luo, M. Rogozia, K. H. Ploog, A. Wacker, and Y. Zhang for their intensive collaboration and stimulating discussions as well as H. Kostial and E. Wiebicke for expert sample processing. Partial support of the Deutsche Forschungsgemeinschaft within the framework of Sfb 296 is gratefully acknowledged. 8. References 1. L. Esaki and R. Tsu, IBM J. Res. Develop. 14, 61 (1970). 2. H. T. Grahn (ed.), Semiconductor Superlattices (World Scientific, Singapore, 1995). 3. A. Wacker and A. P. Jauho, Phys. Rev. Lett. 80, 369 (1998). 4. A. Wacker, in Theory of Transport Properties of Semiconductor Nanostructures, edited by E. Scholl (Chapman and Hall, London, 1998), Chap. 10. 5. H. T. Grahn, in Hot Electrons in Semiconductors, Physics and Devices, edited by N. Balkan (Clarendon Press, Oxford, 1998), pp. 357-381. 6. L. L. Bonilla, J. Galan, J. A. Cuesta, F. C. Martinez, and J. M. Molera, Phys. Rev. B 50, 8644 (1994). 7. L. L. Bonilla, in Nonlinear Dynamics and Pattern Formation in Semiconductors and Devices, edited by F.-J. Niedernostheide (Springer-Verlag, Berlin, 1995), Chap. 1. 8. J. Kastrup, R. Hey, K. H. Ploog, H. T. Grahn, L. L. Bonilla, M. Kindelan, M. Moscoso, A. Wacker, and J. Galan, Phys. Rev. B 55, 2476 (1997). 9. H. Grahn, J. Kastrup, K. Ploog, L. Bonilla, J. Galan, M. Kindelan, and M. Moscoso, Jpn. J. Appl. Phys. 34, 4526 (1995). 10. J. Kastrup, R. Klann, H. T. Grahn, K. Ploog, L. L. Bonilla, J. Galan, M. Kindelan, M. Moscoso, and R. Merlin, Phys. Rev. B 52, 13761 (1995). 11. H. T. Grahn, J. Kastrup, R. Klann, K. H. Ploog, and H. Asai, in Proceed, of the 23rd International Conference on the Physics of Semiconductors, edited by M. Schemer and R. Zimmermann (World Scientific, Singapore, 1996), p. 1671. 12. Y. Zhang, R. Klann, K. H. Ploog, and H. T. Grahn, Appl. Phys. Lett. 69, 1116 (1996).
27
13. J. W. Kantelhardt, H. T. Grahn, K. H. Ploog, M. Moscoso, A. Perales, and L. L. Bonilla, Phys. Status Solidi B 204, 500 (1997). 14. M. Hosoda, H. Mimura, N. Ohtani, K. Tominaga, T. Watanabe, K. Fujiwara, and H. T. Grahn, Appl. Phys. Lett. 69, 500 (1996). 15. N. Ohtani, M. Hosoda, and H. T. Grahn, Appl. Phys. Lett. 70, 375 (1997). 16. N. Ohtani, N. Egami, K. Fujiwara, and H. T. Grahn, Solid-State Electron. 42, 1509 (1998). 17. N. Ohtani, N. Egami, H. T. Grahn, K. H. Ploog, and L. L. Bonilla, Phys. Rev. B 58, R7528 (1998). 18. N. Ohtani, N. Egami, H. T. Grahn, and K. H. Ploog, Physica B 249-251, 878 (1998). 19. K. J. Luo, S. W. Teitsworth, H. Kostial, H. T. Grahn, and N. Ohtani, Appl. Phys. Lett. 74, 3845 (1999). 20. K. J. Luo, S. W. Teitsworth, M. Rogozia, H. T. Grahn, L. L. Bonilla, J. Galan, and N. Ohtani, this volume. 21. O. M. Bulashenko and L. L. Bonilla, Phys. Rev. B 52, 7849 (1995). 22. O. M. Bulashenko, M. J. Garcia, and L. L. Bonilla, Phys. Rev. B 53, 10 008 (1996). 23. Y. Zhang, J. Kastrup, R. Klann, K. H. Ploog, and H. T. Grahn, Phys. Rev. Lett. 77, 3001 (1996). 24. K. J. Luo, H. T. Grahn, K. H. Ploog, and L. L. Bonilla, Phys. Rev. Lett. 81, 1290 (1998). 25. O. M. Bulashenko, K. J. Luo, H. T. Grahn, K. H. Ploog, and L. L. Bonilla, Phys. Rev. B 60, 15. Aug. (1999). 26. Y. Zhang, R. Klann, H. T. Grahn, and K. H. Ploog, Superlattices Microstruct. 21, 565 (1997).
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S P A T I O T E M P O R A L CHAOS IN YTTRIUM I R O N G A R N E T FILMS C.L. G O O D R I D G E , T.L. C A R R O L L , L.M. P E C O R A , and F.J. R A C H F O R D Code 6345, Naval Research Laboratory, Washington, DC 20375 ABSTRACT We describe results from an experiment designed to study the spatiotemporal dynamics of spin wave states in thin films of Yttrium Iron Garnet (YIG). The states of interest are produced by aligning the atomic spins with a DC magnetic field while simultaneously exciting the spins with a RF magnetic field at a resonance frequency. Periodic and chaotic modulations of the spin wave oscillations can occur when the driven spin wave modes interact with half-frequency spin waves. We use a pair of probes to detect the magnetic state of the film at two spatially separated positions on the film and then use both linear and nonlinear analysis techniques to investigate the relationship between the magnetic state at those positions. Our results indicate that periodic states (typically lower power states) are more strongly correlated than more complicated higher power states. We have also determined that there are nonlinear as well as linear components to the relationship between the two signals.
Introduction The study of nonlinear dynamics has become more important with new applications occurring in disciplines ranging from biology to condensed matter physics. In addition, advances in nonlinear analysis techniques have allowed researchers to gain more insight into previously studied nonlinear systems. One example of a condensed matter system that exhibits nonlinear and chaotic properties is auto-oscillations in Yttrium Iron Garnet (YIG) films. YIG is a technologically useful ferrimagnetic material with applications in microwave devices such as limiters, resonators, and filters1. The nonlinear properties of the spin wave dynamics of YIG have been studied and applied for over half a century2. More recently, aspects of the nonlinear dynamics of this system, including both control and synchronization of the chaotic oscillations in YIG, have been studied3-4. Auto-oscillations are low frequency
RF modulations of ferromagnetic spin waves and have been observed in both small spheres and thin films of YIG. When a YIG film is placed in a saturating DC magnetic field, the atomic spins initially align and precess around the direction of the DC field until damped out. If A/-i r- u
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an AC field at a resonance mode of the film (set by the film dimensions and the DC field) is applied perpendicular to the DC field (figure 1), the spins will continue to resonantly precess around the DC field direction at that frequency5"7. Modulations in the phase of 29 across the film surface until reflected at the the coupled spins lead to spin waves traveling
30 boundaries. Standing waves corresponding to the modes of the film result, which initially may be approximated as linear modes. These modes are coupled to half-frequency modes of initially negligible amplitude. However, above a threshold power (the Suhl Instability), these modes begin to drain power from the linear modes. The nonlinearities become large as these half-frequency modes grow in amplitude, causing the low frequency kHz modulation (auto-oscillations) of the GHz spin wave states. Previous researchers have studied spin wave dynamics in a variety of YIG structures. Chaotic transients and attractors produced in YIG spheres and films have been observed and characterized 8 - 1 0 . Other experiments on YIG films investigated mode interactions in these films
Coaxial Probes YIG Film GGG Substrate
0.72 cm
Figure 2: Coaxial probes detect the magnetic moment of the YIG film at two positions.
as well as the effects of using two RF driving frequencies on the Suhl instability 1 U 2 . These earlier experiments were concerned with the global dynamics of the YIG samples and analyzed the temporal dynamics of the auto-oscillations. In these experiments, we study the local dynamics to investigate the spatial variation across the YIG surface.
RF Sweeper
Waveguide Sample Probes RF Amplifiers Diode Detectors Preamplifiers
Figure 3: A diagram of the apparatus used in these experiments.
Experiment Our sample is a rectangular film cut from a single crystal of YIG grown by liquid-phase epitaxy on a gadolinium-gallium-garnet substrate. The film has dimensions 0.85 x 0.72 cm 2 and is 37 microns thick. Auto-oscillations are detected by using two probes next to the film. The probes are constructed by connecting the inner conductor of OS-80 coaxial cable to the outer conductor, forming a small pickup coil. The probes are aligned as shown in figure 2. A diagram of the experimental system is shown in figure 3. The GHz spin wave signals
31
were amplified using low noise Miteq AFS3 microwave amplifiers (providing 35-36 dBm amplification). The experimental parameters are listed in table 1. RF Frequencies RF (Source) Power DC Field Number of Samples Sampling Rate
a. 480.7 G, 3.0802 GHz, 1.75 dBm -i 4 2i-
-2 4
L-L.
0 40 80 120 b. 480.5 G, 3.0547 GHz, 6 dBm
1000 2000 c. 480.7 G, 3.0802 GHz,-1.75 dBm
0 1000 2000 Figure 4: Three examples of time series data observed in these experiments. The state parameters are listed above the plots. Scaled Voltage is plotted against time (each count = 2 (is). The plots are offset to facilitate viewing.
2.7 - 3.3 GHz -20 - +20 dBm 450-515 G 65536 Points 625000 Samples/S
The kHz auto-oscillation modulation is detected using Shockey diode detectors. The signals then are amplified using a Stanford Research Systems 560 analog amplifier and a EG&G PARC 113 analog amplifier and digitized into a computer using a National Instruments I/O board. The DC field was controlled to within 0.01 G by a Varian Fieldial Regulator and measured with a Lakeshore 450 Gaussmeter. The RF excitation power was supplied by a HP 8341 Synthesized Sweeper. The detected RF signals had strengths from -60 to -30 dBm and the modulations had a frequency range of 0.5-200 kHz. Data collection was automated using LabView. Autooscillation states were generated by fixing both the DC field and the RF frequency and varying the RF power. Each data set consists of a time series of the voltage signal from each probe. We analyzed the data using both linear and nonlinear analysis techniques. We first use crosscorrelation to determine the linear relationship between the two signals. We then use a statistic developed by Pecora13 to investigate the nonlinear aspects of the relationship between the two time series.
32
Analysis A variety of different autooscillations can be observed in the parameter range studied. The states initially appear periodic at low powers. As the applied power is increased, the auto-oscillations lose structure and may become chaotic. However, at certain 0.0 1.0 2.0 3.0 parameters, both chaotic and periodic Figure 5: Maximum CrossCorrelation between windows can be observed as power is the time series plotted against applied RF power. varied. Figure 4 shows three examples These states were generated at DC Field 460.7 G and RF Frequency 3.0004 GHz. of the types of states that have been observed. Two of the states appear to be periodic and strongly correlated. The other state appears chaotic and there is much less correlation between the two signals. Figure 5 shows the linear correlation as a function of excitation power for a power sweep at 460.7 G and 3.0004 GHz. The initial increase with power may be due to an increase in the signal strength as the oscillations evolve into periodic states. The linear correlation drops off rapidly at powers above about 1.6 dBm. This reduction in linear correlation coincides with an evolution from periodic to more complicated behaviors. Nonlinear Analysis The goal of this analysis is to measure the predictability between the two time series using nonlinear techniques. The predictability is defined as a measure of how well we can determine the value of a point on one time series, Y;, given its corresponding point on the other time series, Xi. This technique can be extended to investigate the nature of the functional relationship between the two time series. It also allows us to test for any dependence beyond linear dependence between the two time series. The procedure is as follows: 1. A delay embedding is performed on both time series to form two n-dimensional attractors (Wi, W2, W3...) and (Ui, U2, U3...). The point W; corresponds to point Uj. The dimension n is the dimension of the completely unfolded attractor determined using a false nearest neighbor routine14. In the data used, n ranges from four or five for lower power states to greater than eight for the more complicated states. 2. We select a target ball of points, Bt, on one attractor (the target attractor). This target ball is centered around a point Wo and has a radius of • • • i Vi-(m— l)j J/i-Toi • • • > J/i—To-(m-l) j
(2)
51 which means that generically the dimension of our vectors has to be 2m = 4d + 2, even though this is typically smaller than the attractor dimension. Please note that in eq. (2) To is given as multiples of the sampling time At. Now we can use the vectors v to reconstruct the equations of motion and to explore the system. As already mentioned the dynamics on the reconstructed manifold is not unique. Fortunately, the information we need in addition is known. It is the values of y between the two time windows in our embedding vectors. It is important to notice that there is one principal problem with time-delay feedback systems. Due to the finiteness of the sampling time At, one can never exactly reconstruct the flow system, but one has to deal with discrete time systems. For ordinary systems this is not a problem at all, since one can always find a map which is exact. This docs not apply to time-delay systems [10]. The reason is that one has to introduce a "dimensional coarse graining" of the infinite dimensional phase space. This obviously leads to a loss of information. Therefore, the reconstruction can never be exact, but only the best possible approximation. The quality of the approximation is determined by the value of At. The smaller At, the better the approximation. Another way to increase the quality of the approximation is to increase the embedding dimension m. It was shown in [10] that such an improvement can be achieved by increasing either the first window, the second window or both windows simultaneously. III. IMPLEMENTATION Typically, given a time series generated by a time-delay feedback system, one neither knows the delay time To nor the dimension of this system. Therefore, the first steps are the estimation of these quantities. In this section we want to present the general procedure to do this, by using a numerical example. This example is a two dimensional generalization of the well studied Mackcy-Glass system [6]. The equations of motion [16] are:
y(t) = -u>2x(t) - f>y{t) .
{ )
Setting a = 0, eq. (3) reduces to the equations for a damped harmonic oscillator, while the original Mackey-Glass equation reduces to a fixed point equation. The nonlinearity is the same in both systems. Figure 3 shows the time series in a two dimensional delay representation. The parameters were set to: a = 3, J1 = 2 and p = 1.5. The delay time was chosen to be To = 10 and the sampling time was At = 0.1. Using the procedure introduced by Farmer [17] the Kaplan-Yorke dimension turns out to be about DKY = 10.
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2
2.5
FIG. 3. A two dimensional representation of the data integrated using eq. (3. The delay time for the embedding is chosen to be the feedback time TQ. First we want to present a procedure to estimate the delay time TO. It is quite similar to the ones for d — 1 systems by Biinner et. al [18,19]. From the last section we know that there exists a low-dimensional manifold in the embedding space if we choose the correct r . For a wrong T this manifold does not exist. This means that if we choose the wrong delay time, the embedding procedure we use has to fail, since the wis are not restricted to a hypersurface, which means we are dealing with an underembedding. The idea is to make an ansatz for the dynamics yi+i = a 0 + avi(r, m) .
(4)
This ansatz (here a local linear one) can only be reasonable if T is chosen correctly. To get an estimate for To we use r as a parameter and fit eq. (4) to the data in the usual least squares sense. We expect the average forecast error of the model to be large if T is chosen wrongly, while wc expect it to be small for r = To, if additionally m is large enough. In other words we use the forecast error
a m)
^ =i—^)—'
(5)
where & is the forecast obtained by eq. (4) and a(y) is the standard deviation of y, as an indicator for the goodness of our estimate of TO. Figure 4 shows the result for our data. The different curves correspond to different embedding dimensions (m = 1,2,3). One clearly sees a pronounced dip in the forecast error at the position of To (again in units of At). Even for m = 1 this dip is visible, though, m = 1 is not sufficient for an embedding.
53
0.1
0.01
I 0.001
0.0001 0
20
40
60
80
100
120
140
T
FIG. 4. One-step forecast errors as a function of the "unknown" delay time r. The different curves correspond to different embedding dimensions. One clearly sees a pronounced drop of the error at the correct delay time To. One also sees from the figure that the forecast error gains a factor of about 2 when m is changed from 2 to 3. This might indicate that m = 2 is not yet sufficient for an embedding. But this is not the case. For this particular model one can show analytically that m = 2 is sufficient, since one can derive the dynamics in the embedding space. It is given by the second order differential equation x(t) = -u>2x(t) - px(t) + u2f(x(t
- T0)) +
df{x
%~T°VX(t ax[t — To)
- TO) ,
(6)
where / is the function on the r.h.s. of eq. (3) that contains the time delay. The reason why m = 3 is better becomes obvious from eq. (6). For a perfect embedding one needs the first derivatives. Since we do not use derivatives, we have to rewrite eq. (6) in a way that fits our embedding procedure. In other word one has to replace the derivatives with finite time differences. In other words, a higher embedding gives a better estimator of the derivatives and thus a better approximation of (6). This is a typical situation for continuous time systems and the reason why we cannot use the one—step forecast error to estimate the minimal embedding dimension. Thus, we have to choose an alternative strategy to determine the minimal embedding dimension. The idea is to check global properties of the time series like e.g. the power spectrum, the scalar distribution or the mutual information. To do so, we make an ansatz for the dj'namics (usually again a local linear one), take an initial condition from the original time series and then iterate a new trajectory, using the model. After having iterated the new time scries, one can compare the above mentioned properties for different m.
54
FIG. 5. Scalar distribution (left panel) and power spectrum (right panel) of the iterated trajectories for m = 2 and m = 3 and for the original data, respectively. It becomes clear from the plots that an embedding dimension of 2 is sufficient to reproduce all features of these quantities. The power spectrum is given in units of the Nyquist frequency. Figure 5 shows the scalar distribution (left panel) and the power spectrum (right panel) for the original time series as well as for the iterated trajectories for m = 2 and m = 3, respectively. One clearly sees that both quantities show that m = 2 is already in nice agreement with the original data. Once wc know all parameters (TO and ra) for the optimal model wc can start a more detailed analysis of the system. Especially, knowing the dynamics in embedding space allows us to estimate the Lyapunov spectrum of the system [20]. To do this we first have to define the appropriate tangent space. This space is defined by the embedding of the manifold we used so far plus all the points between the actual state and the delayed state, which spans the interval [—To : 0]. Again, this is not the full space of the original system, but only a "coarse grained" one due to the finiteness of At. So we can not expect to compute all Lyapunov exponents, but only a finite subset. Though not rigorously proven, it turns out in all numerical examples that this set of Lyapunov exponents is invariant under the decrease of At, for At small enough. That means that, if we compute N Lyapunov exponents, these exponents coincide with the largest N exponents of the full spectrum [17]. The vectors we use for the construction of the Jacobians are •4 = (Vn, • • • , 2M-Ti,-(m-l)) ,
(7)
and with the local linear ansatz from eq. (4) the Jacobians look like
It has a quite simple structure: The first row contains the coefficients from the ansatz. All other rows just define time shifts of the components of z , as it is typical for spaces defined by time delay embedding vectors. This Jacobian has to be iterated in tangent space to obtain the Lyapunov exponents. Figure 6 shows the results for the first 50 exponents obtained from a time series of length 50000. Also shown in the figure are the first 50 exponents which we obtained directly from the equations
55 of motion using the scheme introduced by Farmer [17]. One sees that the two curves agree quite nicely. 0.05
o -0.05
^-
-0.1 -0.15 -0.2 -0.25
5
10
15
20
25 i
30
35
40
45
50
FIG. 6. The first 50 Lyapunov exponents of the systems. The solid line shows the results obtained from the model directly, while the dashed line shows the results obtained from the data. Let us now suppose that we are dealing with a system which equations of motion do not depend on the delay time To explicitely. This means, the delay time enters the equations of motions only through the delayed variable xn{t — TQ), but not directly. Consequently, the geometry of the manifold, we are willing to reconstruct, does not depend on the delay time. Or in other words, if we reconstruct the manifold for a given TO, we know it for all other delay times. Thus, we can investigate the behaviour of the system for different delay times, by just having one data set. Of course, the measure on the manifold docs depend on the delay time. The hope is that, if we have access to data that stems from a high dimensional regime (large To), the support of the measure for smaller TS is a subset of that of the large delay time, at least. 1 0.8 0.6 0.4
*
0
-0.2 •0.4 -0.6 -0.8 -0.8 -0.6 -0.4 -0.2 0
0.2 0.4 0.6 0.8
1
FIG. 7. Two-dimensional representation of data produced from cq. (9. The delay time was set to TQ = 20. To avoid problems we always encounter with time continues systems, we demon-
56
strate the idea using a scalar, time discrete model given by by2n_Ta .
yn+1 = l-ayl-
(9)
The parameters were chosen to be a — 1.2 and 6 = 0.6. Figure 7 shows a delay plot for a delay time T0 = 20. For these parameters the Kaplan-Yorke dimension turns out to be DKY « 8. We now fit a model to the data using a trajectory of 20000 points. After having obtained the model, we use it to iterate a trajectory not for To = 20, but for TQ = 5, as shown in the right plot of fig. 8.
0.8 -
0.8 •
0.6 -
0.6 •
0.4 •
0.4
2
/ ° ' o •
-0.2
-0.2 • -0.4 -„.„ —. . -0.6 -0.4 -0.2
-0.4
. 0
. ' • • 0.2 0.4 0.6 0.8
• 1
-0.6 I ' ' -0.6 -0.4 -0.2
•0
yi
' ' ' ' 0.2 0.4 0.6 0.8
1
y>
FIG. 8. Two-dimensional representation of data produced from eq. (9. The left panel shows data obtained directly from the model equations using TO = 5. The right panel shows the data for the same delay, but obtained from the data shown in Fig. 7 iterated with TO = 5. The left panel in fig. 8 shows the data obtained directly from the model. One sees that the attractors coincide nicely in this two-dimensional representation. Of course, this is rather a qualitative coincidence than a quantitative one. To check the quantitative coincidence we show in fig. 9 the Lyapunov specta for both data sets. 0.05
o -0.05 -0.1 -0.15 -0.2 -0.25 -0.3
1
2
3
4
5
6
FIG. 9. Lyapunov spectra obtained from the iterated trajectory (dashed line) and the model data (solid line), respectively. From the figure one sees that the results agree nicely. Calculating the Kaplan-
57 Yorke dimension one finds that the difference between both is below 2 percent (DKY «* 2.2). IV. DISCUSSION We presented a method to analyse time-delay feedback systems. By introducing a new embedding method we can study these systems even in the case that the dimension of the chaotic attractor is very high. In principle there is no restriction to the dimension of the attractor. The reason is that we do not try to analyse the chaotic attractor but a manifold that is defined by the equations of motion. The dimension of this manifold does not depend on the attractor dimension. Its geometry is invariant under the change of r 0 , at least if the equations of motion do not explicitely depend on T 0 . The systems we studied so far are given by eq. (1). We required that only a single variable with a single and fixed delay time was fed back. This is of course not the most general class of systems. Furthermore, we have supposed that the variable measured is the one which is fed back. Of course, there could be systems where this is not possible. So the question is, whether it is possible to extend our method to more general situations. Let us start with the case where we have more than one delay time. In principle this is possible. Using the same arguments as in sec. II it is obvious that we need a window for each delay time, at least, if the windows do not overlap. This means that we loose the advantage that the method works in fairly low-dimensional spaces with an increasing number of delays. Therefore, more than one delay time is not a theoretical problem, but a practical one since we have to reconstruct a higher dimensional manifold, which means we need more data. The situation changes dramatically if wc suppose the system to have a infinite number of delay times. For example the equations of motion could contain a whole interval of delay times. If this window is sufficiently small, so that it can be replaced by an average delay time plus some small fluctuations, the method may work approximately. Approximately in the sense that we do no longer have a manifold in a strict sense, but it is smeared out instead. But if the interval becomes larger this approximation becomes worse, so that the method finally has to fail. The next question is, what happens if we do not measure the delayed variable but a different one? Generally, the arguments given for the input-output systems are not applicable. In our case, they still work. The main difference is that we have to reconstruct the unmeasured delayed variable from the data wc have at hand. Wc could show that this is possible, but we have to pay the price of doubling the embedding dimension, generically [10]. The same arguments hold for the case that we have more than one variable fed back. At least one of the components is not measured and we have to use a higher embedding dimension. The results presented here were obtained by means of numerical examples. The reason to present these examples was to avoid difficulties inherent in the analysis of
58 real data. There one has to handle problems like noise in the data or nonstationaxity. Meanwhile we were able to successfully apply the methods to a experimental system, a C 0 2 laser experiment [21] performed at the National institute of Optics in Florence, Italy. The results of this analysis will presented elsewhere [22].
H. D. I. Abarbanel, R. Brown, J. L. Sidorowich, and L. S. Tsimring, Rev. Mod. Phys. 65, 1331 (1993). H. Kantz and T. Schreiber, Nonlinear time series analysis (Cambridge University Press, Cambridge, UK, 1997). T. Schreiber, Phys. Rep. 308, 1 (1998) J.-P. Eckmann, D. Ruelle, Physica D 56, 185 (1992) E. Olbrich and H. Kantz, Phys. Lett. A 232, 63 (1997) M.C. Mackey and L. Glass, Science 197, 287 (1977) R. Lang and K. Kobayashi, IEEE J. Quantumn Electron. QE-16, 347 (1980) J.K. Hale and S.M.V. Lunel, Introduction to functional differential equations (Springer, New York, Heidelberg, 1993) R. Hegger, M.J. Bunner, H. Kantz, and A. Giaquinta, Phys. Rev. Lett. 81, 558 (1998) M.J. Biinncr, M. Ciofini, A. Giaquinta, R. Hegger, H. Kantz, R. Mcucci, and A. Politi Reconstruction of systems with delayed feedback: (I) Theory, submitted for publication (to be found on the eprint server: xyz.lanl.gov, Ref. No.: chao-dyn/9907020) N. Packard, J. Crutchfield, and J. Farmer, R. Shaw, Phys. Rev. Lett. 45, 712 (1980) F. Takens, in Dynamical Systems and Turbulence, Warwick 1980, Lecture Nots in Mathematics, edited by D.A. Rand and L.-S. Young (Springer, Berlin, Heidelberg, 1980), Vol. 898, 366 T. Sauer, J.A. Yorke, and M. Casdagli, J. Stat. Phys. 65, 599 (1991) M. Casdagli, in Nonlinear Modeling and Forecasting, SFI Studies in the Sciences of Complexity (Addison-Wesley, Reading, MA, 1992) J. Stark, D.S. Broomhead, M.E. Davies, and J. Huke, Nonlin. Analysis, Methods & Applications 30, 5303 (1997) M.J. Biinner, Th. Meyer, A. Kittel, and J. Parisi, Phys. Rev. E 56, 5083 (1997) J.D. Farmer, Physica D 4, 366 (1982) M.J. Biinncr, M. Popp, Th. Meyer, A. Kittel, U. Rau, J. Parisi, Phys. Lett. A 211, 345 (1996) H. Voss and J. Kurths, Phys. Lett. A 234, 336 (1997) R. Hegger, Estimating the Lyapunov spectrum of time delay feedback systems frvm scalar time series, to appear in Phys. Rev. E (1999) F.T. Arecchi and W. Gadomski and R. Meucci, Phys. Rev. A 34, 1617 (1986) M.J. Bunner, M. Ciofini, A. Giaquinta, R. Hegger, H. Kantz, R. Meucci, and A. Politi Reconstruction of systems with delayed feedback: (II) Applications, in preparation
CHAOS CONTROL IN FAST SYSTEMS USING OCCASIONAL FEEDBACK Ned J. Corron Dynetics, Inc., P. 0. Box 5500, Huntsville, AL 35814, USA Krishna Myneni, Thomas A. B a n SAIC, 6725 Odyssey Drive, Huntsville, AL 35806 Shawn D. Pethel U. S. Army AMCOM, AMSAM-RD-WS-ST, Redstone Arsenal, AL 35898
ABSTRACT A need to stabilize diode laser systems motivates the development of chaos control techniques for very fast systems, i.e., natural frequencies of 3 GHz or faster. In this paper, we present a new occasional proportional feedback control that is very simple to implement at high speeds yet retains capability to capture unstable periodic orbits using small perturbations. Our approach uses a pulsewidth-modulated control signal derived from the passage of the system state through a prescribed window. In operation, the average control power is determined by the window transit time and, for a properly placed window, is related to the deviation of the system state from the targeted orbit. Practical implementation requires just a few comparators and a single logic gate. Experimental control is demonstrated for a 1-kHz, piecewise-linear Rossler circuit and a 19-MHz, Colpitt's oscillator. For the latter, a fast electronic controller, with bandwidth exceeding 200 MHz and latency below 5 ns, was built using commercially available components and a conventional printed circuit board. Ultimately, we believe this control technique can achieve diode-laser speeds using integrated or hybrid circuit implementations. The paper concludes with a discussion of an elegantly simple, analogous control scheme for mechanical systems.
1. Introduction Chaos control offers an intriguing approach to quelling to instabilities in diode laser systems with optical feedback from external reflections. The resulting chaotic behavior, which is modeled by the Lang-Kobayashi equations [1], is characterized by irregularly spaced, sub-nanosecond pulses. These pulses impart a very large spectral linewidth to the laser and limit the modulation bandwidth of these devices. The objective of chaos control is to use small perturbations to stabilize unstable periodic orbits (UPO), which are'abundant in the dynamics of a chaotic attractor. Stabilizing a specific UPO in a chaotic diode laser system will dramatically sharpen the spectral line and improve the efficiency of the laser. The initial description of chaos control is due to Ott, Grebogi, and Yorke (OGY) [2]. The OGY algorithm is a mathematical prescription for perturbing the system dynamics and placing the system state on a stable manifold of a targeted UPO. This is 59
60 done by sampling the system on a return map and exploiting local linear behavior near the fixed point of the UPO. As a result, a small control signal is applied only occasionally; that is, the control is applied only when the system returns to the map within a prescribed window. Otherwise, the system is not perturbed. The OGY algorithm, which was experimentally verified by Ditto et al. [3], requires several vector calculations to generate the control signal. Hence, OGY is practical for only slow systems. Hunt successfully demonstrated a simpler variant of OGY, called occasional proportional feedback (OPF), that enabled control in electronic circuits operating upwards of 105 Hz [4]. However, it is generally believed that for very fast systems, a continuous feedback, such as described by Pyragas [5], is required. In fact, using extended time-delay autosynchronization (ETDAS), the chaotic dynamics of a diode resonator driven at 10.1 MHz were stabilized [6]. Until now, this is the fastest system reported to be stabilized using chaos control. In this paper, a new chaos control algorithm is presented that refutes the assumption that continuous feedback is required for controlling very fast systems. This new technique, called transit-time pulsewidth modulation feedback (TPF) [7], uses a pulsewidth-modulated control signal derived from the transit-time of the system state through a prescribed window. Although pulsewidth modulation has been considered previously [8,9], the present implementation is significant since its simplicity enables applications demanding very high frequency response and minimal latency to achieve effective chaos control.
2. Transit-Time Pulsewidth Modulation Feedback (TPF) Control Consider a physical system described by the equation
x = f(x;n)
(1)
where x is a vector of system states, / is the vector field, and fi is an accessible system parameter. A control scheme applies a perturbation to the system parameter to stabilize a targeted state, such as an UPO. For example, in OPF, the perturbation is |
0
otherwise
where x\ is the intersection of the system trajectory with a specified surface of section, xr is the intersection of the targeted state with the same section, a is a fixed gain, A(T) is a fixed pulse of duration T, and e defines a neighborhood about xT for which the control is active. In contrast, the TPF control perturbation is fa 8u = , the memory parameter R and the frequency deviation $ = ft — TT/T at K^• i, which does not contain any system parameter. Experimental data obtained from the control of an unstable period-4 orbit are presented in Fig.
72
40
«
60
Figure 4: Dependence of Floquet exponent on control amplitude.
2000
0
1000
v [kHz]
Figure 5: Power spectra of system output for different control amplitudes.
73
AD A
OQ ..•
A / 6 •,.-;;'-'' A,.--'' o A--' •&
0
10
20
p-'X'
o--g.--
• g' g
30
40 K
50
60
70
Figure 6: Stability range in the K-R parameter plane for three values of the driving amplitude: • 0.8V, o 1.1V, A 3.5V. Full/open symbols correspond to the flip/Hopf boundary. Solid/dashed lines indicate the analytical result. 7. Keeping in mind that the corresponding analytical curves are obtained without any fit parameter the coincidence is remarkable. Hence, the essential qualitative and several quantitative features which determine the stability domain for time-delayed feedback control can already be described by our first-order analysis. 3.2
Rossler-type
electronic circuit
Further features of time-delayed feedback control were probed on a Rossler-type nonlinear circuit, consisting of several operational amplifiers with associated feedback components (cf. Fig. 8). The nonlinearity is provided by the diodes. The voltages probed at x,y,z can be considered as the degrees of freedom in our experiment. At fx,fy,fz external signals can be fed into the system for control purpose. Typical frequencies of the circuit are about 600kHz. Without control the system undergoes a period-doubling cascade to chaos on variation of the resistance Ry, ending up in a chaotic attractor. The following measurements have been performed at Ry = llOfi were the chaotic attractor contains an unstable period-1 orbit with period T = 1.656/xs and Floquet frequency u> = TT/T. The delay time r was adjusted according to the orbit's period T. Here, we only present results obtained for memory parameter R = 0. Our feedback scheme consisted of coupling the voltage at z via the control device to fz. Apart from studying methods of fine adjustment of the delay time r in autonomous systems [9] we used this circuit to investigate the influence of control loop latency S on the efficiency of control. This additional delay which acts on the control force leads
74
0.75 K (fl) / K (ho>
Figure 7: Ratio of critical control amplitudes in dependence on the frequency deviation $ at the Hopf instability for several values of R. Symbols are results of the electronic circuit experiment, lines display the analytical expression: R = 0 ( • , solid line), R = 0.2 (o, dashed line), and R = 0.5 (A, dotted line).
Figure 8: Experimental setup of the nonlinear electronic circuit without the time-delayed feedback device. Experiments have been performed at Rv = UOU.
75
T
i
i
I
1
i
i
J
I.
i
r
1
_L
0.4
K 0.3
0.2
J
0
50
100
150
200
5[ns] Figure 9: Dependence of control interval on control loop latency, • : KW>(5), O: Kvl\8). The gray-shaded region is not accessible in our experiments due to the intrinsic latency 5Q. The lines are fits of the analytical result to the experimental data. to a shrinking control interval by shifting the frequency splitting point Kopt. This phenomenom has been analysed analytically in [10] and it was possible to determine a critical value 8C = T ( 1 — Ar/2)/(Ar) where stabilisation is no longer achieved. Moreover we note that there always exists a (^-interval within [nr, (n + l)r] where control will fail. The latency effect was realized by including an additional delay line between control device and feedback input. The control loop without additional delay line had a latency 0.3). Note that the orbits stabilized by the control are not the periodic orbits of the original system. It is still an interesting problem to find if the controlled orbits are close to the original periodic orbits and if it is stable. The test is easily performed by withdrawing the control after the system has settled into a controlled periodic orbit. We find that, for the lp, 4p and 8p orbits the system moves on to a quasiperiodic orbit near the lp, 4p and 8p elliptic orbits of the Hamiltonian system, respectively. This implies that the control only stabilizes elliptic orbits. The elliptic periodic orbits are more easily stabilized than hyperbolic ones, because for weak damping the correction of Lyapunov exponents of an elliptic orbit is not so large, while it is difficult to convert a large positive Lyapunov exponent of a hyperbolic periodic orbit into a negative value. Fig.2(a) is the trajectory (z,i) of the recovered system starting from a controlled 4p orbit , and (b) is the
84 corresponding power spectrum. The expansion of the spectrum line shows that the system moves on a quasiperiodic orbit. The dynamic features depend on the control intensity. Fig.3 is a bifurcation diagram of the system illustrating the average kinetic energy versus control intensity E. For small e " y —=^ U C2
L,C,
TJL2I^2
•JL2L2 ILC,
--Ur
•uwr C3 '
Consider the features of various oscillation modes of the oscillator model described by Eqs. 6. The value of parameter T has a strong effect on the conditions of the signal propagation along the oscillator feedback loop and determines its entire pass band. With T« 1, the lowpass i^iCi-filter cutoff frequency is high (oiutoj?-= 1/7), the resonant frequencies of R2L2C2- and i?3Z,3C3-filters fit into the .RiG-filter pass band and are passed through it with practically the same gain. On the other hand, in the case of large values of T these resonant frequencies practically don't fit in the .RiG-filter pass band. In the range of medium T, the resonant frequencies of both TiLC-filters fall on the cutoff of the i?iCi-filter amplitude-frequency response, but they are passed with different gains. In this range of the values of T the system dynamics is considerably determined by twofrequency oscillations.
Fig. 3. Bifurcation diagram (maximum values y as a function of gain M),
124 As follows from an analysis of the effect of a on the oscillator dynamics, if a is near one, the amplitude-frequency responses of the RLC-&\ters greatly overlap, and the resonant frequency of the bandpass RiLiCi-filter fits in the pass band of the lowpass iy^CV-filter. Inversely, with 45 Hz with T taken as analogous to r. Clearly the CCM, model as so far specified, cannot reproduce the experimental phase diagram Fig. 1 since that diagram is two dimensional (T and fo), and our CCM model so far has only one parameter, r. Thus, we seek to introduce a new parameter into the model. To do this we choose to introduce a second length scale Ac -In I kc in addition to the length scale \ . Thus, the ratio of these two length scales (kjko) will appear as an additional parameter of the CCM model. We introduce this additional length scale by the following, somewhat arbitrary, modification of our original choice for i ( k ) , L(k) = sgn{kc-\k\)expy(k)
,
(6)
where y(k) is still given by (5). (We have also tried another choice for L incorporating the additional scale Xc, and obtained qualitatively similar results.) Figure 3 shows the phase diagram for our CCM model (4)-(6). Clearly, there is very good qualitative agreement with the experimental phase diagram, Fig. 1. Figures 4 show various patterned states produced by our model: 4(a) fo/2 stripes, 4(b) fo/2 squares, 4(c) competing f0/2 stripes and squares, 4(d) fo/2 hexagons, 4(e) fo/2 flat states separated by a kink, 4(f) f0/2 flat states separated by a "decorated" kink (the existence of experimental decorated kinks is reported in Ref. 7), 4(g) a pattern disordered in space and time. 4. Modeling the Oscillon Phenomenon Using the CCM Approach So far, the model has done well modeling the narrower width layers (compare Figs. 1 and 3). The occurrence of oscillons in thicker layers, however, is found to be accompanied by hysteresis, and this is conjectured to be crucial. The map M(x,r) we have so far used (i.e., Eq. (4)) has a nonhysteretic period doubling (Fig. 5(a)). Thus to test the role of the hysteresis that occurs for thicker layers, we use a map with a hysteretic period doubling (Fig. 5(b)). M(x, r) = -(rx +
x2)exp-(x212)
(7)
147 As detailed in Ref 4 this modification of the model immediately yields oscillon phenomena similar to the experimental observations. This supports the hypothesis that hysteresis is crucial for oscillons. This illustrates one of the advantages of CCM models for this kind of system, namely the ease with which different effects can be tested by transparently incorporating and removing them in the model. 5. Conclusions The fact that only very general, physics-independent inputs to the model (period doubling, two spatial scales, hysteresis) are sufficient to reproduce experimental results on vibrated granular layers, suggests that phenomenology similar to that observed in the granular experiments should also occur in physically different systems that are periodically driven. We suggest that a good place to look for this is in systems capable of pattern formation, but operated in a parameter regime where only uniform states occur (i.e., no patterns). The presence of only uniform states suggests strong dissipation, as in granular media. If such a system is then periodically driven, period doubling of the homogeneous state might be expected, and hence phenomena similar to those in Refs. 4-6 could be expected. As possible candidate systems we mention the chemical system periodically forced by laser light studied in Ref. 8, and the ionized gas discharges discussed by Purwins et al.9 where, in the latter case, the driving might be accomplished by adding a sinusoidal component to the dc voltage applied across the plasma layer. Finally, we note that other useful related modeling works on the experiments of Umbanhowar et al.4"6 have been done by other groups using a variety of approaches10. The work of E.O. was supported by the Office of Naval Research (Physics) and that of S.C.V. by NSF DMR 9415604. 6. References 1. E.g., M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 854 (1993). 2. E.g., D. K. Umberger, et al., Phys. Rev. A 39, 4835 (1989). 3. E.g., K. Kaneko, Chaos 2, 279 (1992). This issue of Chaos focuses on CML's. 4. S. C. Venkataramani and E. Ott, Phys. Rev. Lett. 80, 3495 (1998). 5. F. Melo, P. B. Umbanhowar, and H. L. Swinney, Phys. Rev. Lett. 75, 3838 (1995). 6. P. B. Umbanhowar, F. Melo, and H. L. Swinney, Nature 382, 793 (1996). 7. P. B. Umbanhowar, F. Melo, and H. L. Swinney, Physica A 249, 1 (1998). 8. V. Petrov, Q. Ouyand, and H. L. Swinney, Nature 388, 655 (1997). 9. H. -G. Purwins, this proceedings. 10. T. Shinbrot, Nature 389, 574 (1997); E. Cerda, F. Melo, and S. Rica, Phys. Rev. Lett. 79, 4570 (1997); L. Tsimring and I. S. Aronson, Phys. Rev. Lett. 79, 213 (1997); H. Sakaguchi and H. R. Brand, J. Phys. II (France) 7, 1325 (1997); C. Bizon, M. D. Shattuck, J. B. Swift, W. D. McCormick, and H. L. Swinney, Phys. Rev. Lett. 80, 57 (1998); D. H. Rothman, Phys. Rev. E 57, R1239; J. Eggers and H. Ricke, Phys. Rev. E 59, 4476(1999).
148
8
* -4
^ „ ^ DISORDERED * * * * * * * . , ^AGON%^ 4 4 * + ' * 4)
7
SQUARES (f/4) STRIPES (f/4)
6
r> «
FLAT WITH KINKS
5
•
•
•
•
•
•
•
>
4
38
•
•
^
•
•
•
•
•
•
•
•
•
^ HEXAGONS ( f ^
•
•
•
SQUARES (f/2) ; 3
1
• • • ] n n •
•
n
STRIPES (f/2)
m A ~ " H
• • • • •
2
i
FLAT
10
30
50
70
•
„ • • > • : •
90
f(Hz)
Figure 1. Experimental phase diagram.
110
149
Figure 2. y(k) versus k.
(a)
Disorder
QO O O O O O O
OOOOOOOOOO
Period 4 Stripes OOOOOOOOOOOOOOOOOOOOOOOO
Period 2 Flat State
ooooooooo aoooooooooooooo Period 2 Hexagons oooooooooooo o o o o Period 2
Period 2 _
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oooooooooo Period 1 Flat State
Coexisting Period 2 Stripes & Squares I
1.0
3.0
2.0
(kc/kj
Figure 3. Phase diagrsim for the CCM model specified by Eqs. (4)-(6).
.
4.0
151
Figure 4. Patterned states produced by the CCM model specified by Eqs. (4>(6).
152
* iJLP
(a) Figure 5. Nonhysteretic (a) and hystcrctic (b) period doublings.
153
*
(b)
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MAGNETICALLY INDUCED SPATIAL-TEMPORAL INSTABILITY IN A FERROFLUID Weili Luo1, Tengda Du1, and Jie Huang2 'Department of Physics, Advanced Materials Processing and Analysis Center, and Center for Drug Discovery and Diagnostics, University of Central Florida, Orlando, FL 32816. 2 Department of Civil and Environmental Engineering, West Virginia University, PO Box 6103,Morgantown, WV 26506-6103 Abstract We discuss a novel magnetic-field-induced instability in ferrofluids. The mechanism behind the instability is a nonuniform magnetic body force, arising from spatial inhomogeneity in the magnetic susceptibility of the fluids through either temperature or particle concentration gradients. Using a simple and innovative technique, we are able to produce a controllable temperature distribution, which generates a concentration gradient via the thermal diffusion of particles. We show that this force leads to new instabilities that have never been studied before, either experimentally or theoretically. Possible route to chaos will be discussed. I. Introduction Most of instabilities studied so far are driven by gravity, surface tension, and ponderomotive force1. Here we introduce a magnetic force into the study on instability phenomena. A nonuniform magnetic field exerts a body force on all classical fluids such as water and artificial fluids such as ferrofluids. This force can drive the fluids to unstable states analogous to the buoyancy driven convection. Ferrofluids studied in this work consist of magnetite particles suspended in nonmagnetic solvents2 as shown in fig. 1. The mean diameter of these particles is 9 nm and each particle is coated with a non-magnetic surfactant layer of 2 nm in thickness to prevent agglomeration. The average magnetic moment is in the order of 104 Bohr magnetons. The typical particle volume fraction is several percents. In the absence of magnetic fields, the whole fluid has a null net moment at the room temperature due to Brownian motion of particles. A ferrofluid placed in a magnetic field is subject to the Kelvin body force per unit volume fm = Uo(M-V)H', which arises from the interaction between the local magnetic field H' within the fluid and the magnetic moments of the particles characterized by the magnetization M (the magnetic moment per unit volume). Here, u« is the permeability of free space. This force tends to move the fluid toward regions of higher magnetic field. The fluid is isotropic and the magnetization satisfies M = x(T,C)H' for smallfield,where 155
156 X is the magnetic susceptibility of the fluid following Curie's law, % cc C/T. Here, T is the temperature and C the particle concentration. In the presence of a uniform external vertical magnetic field H, the internal magnetic field in a laterally unbounded horizontal layer of the fluid has the form H' = H/(l+x). Since Vx H'=0, the Kelvin body force follows as
m
2
VH
" (T777 I T
-^)>
0)
where H is the magnitude of H and H' the magnitude of H'. Thus both temperature and concentration gradients can render this force spatially nonuniform even if the external field is uniform. This inhomogeneous body force can promote instability in the fluid in a manner similar to the buoyancy-driven instability in Rayleigh-Benard (RB) convection. However, unlike RB convection in which gravity is a constant, the magnitude of the force in equation (1) can be easily altered by changing the applied field. More importantly, the vector field introduces a vector control parameter whose components have distinct roles, providing an opportunity to study their different effects.
Solvent
Fe
3 °4
Surfactant
Fig. 1 A typical fcrrofluid. II. Results and Discussion We study magnetic-field-induced instabilities in ferrofluids by an innovative technique3 in which the instabilities are induced by the Kelvin force in a horizontal layer of ferrofluids when a laser beam is focused on the layer from below. The schematic experimental set-up is illustrated in fig. 2. A 7 mW He-Ne laser beam is focused
157
normally on a thin layer of ferrofluid (100 fim) by a lens. The sample is sandwiched between two parallel glass-plates. The far-field diffraction patterns are detected by a CCD camera. In zero applied field, when a laser beam with a Gaussian profile is focused on the sample cell, a temperature gradient against the radial direction is established in the system due to the fluid's large absorption coefficient of 550/cm and the increased light intensity from the focusing lens. This thermal gradient induces a particle concentration gradient via the thermal diffusion or the Soret effect4. These radial gradients of concentration and temperature yield a radial profile in the refractive index of the fluids, which in turn produces diffraction patterns at far field above the sample such as multiple concentric ring shown in fig. 3 (a). The number of rings depends on the light intensity, the focal length of the lens, the concentration of particles, and the thickness of the sample layer. We found, by numerical calculation, that the temperature difference is about 15 AT between the beam axis and the beam edge5, which leads to a positive concentration gradient of particles that is found to be 70% on the optical axis.
Fig. 2 The experimental setup: 1- He-Ne laser. 2 - mirror. 3 - lens for focusing the incident beam. 4 - coil for producing magnetic field parallel to laser beam. 5 - sample cell. 6 - Lens. 7 - screen. 8 - CCD camera. 9 - video recorder. 10 - computer. 11 monitor. 12 - Gauss meter. 13 - power supply.
158
(a)
(b)
(c) Fig. 3 (a) Multiple concentric rings in zero magnetic field; (b) Circular symmetry is broken when the applied field H just above the critical value. He; (c) One of the polygon shapes in H > He.
159
In the presence of a magnetic field, the circular symmetry of the ring is broken when the applied field is larger than a threshold, (fig. 3 (b)), then the time dependence sets in and the instability starts. The temporal patterns are changing between polygons such as triangle6, tetragon, and pentagon as illustrated infig.3 (c). In order to understand the physics for this instability, we consider a twodimensional fluid flow. Although the temperature continues to decrease outside the beam, it levels off at several (4-5) beam widths; the corresponding temperature difference is about 40 K. So the temperature has a significant change only in a range of several beam widths from the beam center. Since the particle-concentration gradient is induced by the thermal gradient, the concentration also significantly varies only in the same range as the temperature. Thus, only within this range, the magnetic Kelvin body force has significant variations. Outside this range, the Kelvin body force is very small because of the small temperature and concentration gradients. Accordingly, it is plausible to assume that the field-induced convective fluid flow, thus the convective rolls only occur within this range. A linear stability analysis should provide a criterion for the onset of these convective rolls. However, the analytical solution for this cylindrical geometry is almost impossible. To obtain a simple analytical criterion for characterizing the instability without losing physical significance, we considered a simplified twodimensional model in which the flow in the sample plane is confined between two laterally unbounded parallel lines with a temperature difference AT = 40 K across them6. Our linear stability analysis for this geometry yields the criterion for the onset of instability. An estimation of the critical field Hc for H yields a value of the same order as observed, confirming our qualitative analysis.
Fig. 4 Configuration of the convective rolls that gives rise to the diffraction pattern infig.3 (c).
160
The axial symmetry of the laser beam and the applied field plus the geometry of the sample (very small thickness) suggest that the axes of convective rolls should be parallel to the beam axis. There must be even number of the rolls because of periodicity and the symmetry. In a stability analysis7 on a ferrofluid in a cylindrical shell with radial magnetic and temperature gradients, Zebib demonstrated that states that are most easily to excite are the ones with six, eight, and ten convective rolls. Fig. 4 illustrates the configuration of the convective rolls responsible to the diffraction pattern with pentagon shape shown in fig. 3 (c). Neighboring rolls with inward radial flow between them leads to a concave side and any two adjacent rolls with outward flow form a corner thus the ten rolls form the pentagon. Similarly, a six-roll state gives rise a triangle and an eight-roll state a tetragon that have been observed as well6'8. III. Conclusions Kevin force can drive a system to instability that is analogous to buoyancy force for RB convection. For ferrofluid, because of large magnetic moments of particles, the critical field required for the bifurcation is several orders of magnitude smaller than ordinary diamagnetic or paramagnetic fluids. The symmetry of the system and the geometry of the sample cell lead to the special configuration of convective rolls that gives rise to polygon shapes in the diffraction patterns that have rarely been studied before. The observations discussed here indicate that the mechanism responsible for the instability could be described by nonlinear coupled equations, thus chaotic behavior is possible if we impose a time dependent force which will drive the system through various dynamical states. IV Acknowledgment We are grateful for Dr. R. E. Rosensweig for providing the sample. This work is partially supported by NSF Young Investigator Award to Luo and NSF DMR.
1
For a review, see M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993) and references therein. 2 For details about magneticfluids,see R. E. Rosensweig, Ferrohydrodynamics (Cambridge University Press, 1985). 3 T. Du, S. Yuan, W. Luo, Appl. Phys. Lett. 65, 1844 (1994); T. Du and W. Luo, Modern Physics Letters B 9, 1643 (1995). 4 C. Ludwig, Sitzungber, Acad. Wiss. Wien 20, 539 (1856); C. Soret, Arch. Sci. Phys. Nat. Geneva 2, 48(1879). 5 T. Du and W. Luo, Appl. Phys. Lett. 72, 274 (1998). 6 Weili Luo, Tengda Du, and Jie Huang, Phys. Rev. Lett. 82, 4134, (1999). 7 A. Zebib, J. Fluid Mech. 321, 121 (1996). 8 Weili Luo, Tengda Du, and Jie Huang, J. Magn. Magn. Mat. 1999.
PATTERN CONTROL WITH SPATIAL PERTURBATIONS IN A WIDE APERTURE LASER
R. MEUCCI, A. LABATE, M. CIOFINI Istituto Nazionale di Ottica, Largo E. Fermi 6, 50125 Florence, Italy PENG-YE WANG Laboratory of Optical Physics, Institute of Physics and Center for Condensed Matter Physics, Chinese Academy of Sciences, P. O. Box 603, Beijing 100080, People's Republic of China
ABSTRACT Pattern selection and stabilization by means of spatial perturbations is experimentally investigated in a wide aperture C02 laser. Thin metallic wires are inserted into the laser cavity to realize the spatial perturbation. The effects of a single wire on the fundamental and annular patterns show that the diffraction induced by the wire plays an important role. By using more wires hexagons are obtained. By changing the cavity detuning we observed the transition from a hexagon to a dodecagon via the doubling of the azimuthal spatial frequency.
1. Introduction The success of different algorithms to control temporal chaos, starting from the pioneering work of Ott, Grebogi and Yorke1, has increased the interest, in the field of nonlinear dynamics, to generalize the control techniques to the _ space-time domain. Control of optical turbulence by means of Fourier plane filtering2 and by generalization of the Pyragas method3 have been proposed4. Recently, Wang et al.5 suggested a nonfeedback method to stabilize, select and track unstable patterns based on weak spatial perturbations exerted on a control parameter of the nonlinear system. In general, the perturbed control parameter (i of the system can be written as5 tt = H)(i + cf(ry),
(l)
where f(r) is the spatial perturbation function. The perturbation function / ( ? ) should be designed to reflect the features of the target pattern. An important advantage of this perturbation method is its experimental feasibility. The early studies concerning the effects of spatial perturbations on laser systems were performed by Rigrod on a He-Ne laser6. In this paper we present the results obtained by inserting thin metallic wires inside the cavity of a highly symmetric CO2 laser. In the case of a single wire it is possible to modify the type of symmetry breaking 161
162
by varying the position of the wire, obtaining stabilization of different spatial patterns7. We also show that by using more wires in different spatial configuration, it is possible to select and stabilize more complex patterns. 2. Experiment The experimental setup consists in a Fabry-Perot laser cavity 700 mm long. The grid that is composed of metallic wires is placed inside the cavity at a distance of 110 mm from the spherical outcoupler mirror of the laser. The laser output patterns were observed by means of an infrared image plate placed at a distance of 400 mm from the outcoupler mirror. Considering our configuration, the observations are made in the farfield region. The observation of the effects of a single wire is the basis to understand the effects of the grids with more complicated geometries. Considering a thin wire with diameter of d=100um and the laser wavelength of A.=10.6um, the Fraunhofer diffraction condition z » 7td / A = 314mm, where z is the longitudinal coordinate, is satisfied at both the mirrors of the laser cavity. The Fraunhofer diffraction pattern of a thin wire can be analytically obtained considering the wire as a one-dimensional rectangular function rect[x\ I d), where xi is the transverse coordinate perpendicular to the wire. The Fourier transform of this function along the xo axis at an observation distance z from the wire is sincydxQ I kz). The width between the first two zeroes is AXQ = 2Az I d .In our case die width of the main lobe is 22 mm on the outcoupler mirror. Since the diameter of the fundamental mode TEMoo on the same mirror is 5 mm, the effect of the wire is a weak spatial modulation. We verified that the optical power reduction of the fundamental mode due to the insertion of the wire outside the optical cavity, is about 4 %. Inside the cavity, the spatial perturbation induces a weak transverse modulation, which selects and stabilizes different laser output pattern depending on the symmetry of the perturbation. First we consider the fundamental mode, obtained with a intracavity diaphragm aperture of 7.6 mm. If the wire crosses the optical axis, the power reduction is about 50% and the resulting output pattern displays two intensity lobes with a central line of zero intensity as shown in Fig. 1. In this case, the circular symmetry of the cavity is broken, but the cavity keeps inverse symmetry. The first axis of inverse symmetry is the axis of the wire and the second is the axis perpendicular to the wire and passing through the center. Due to the inverse symmetry, the laser field on both axes is mapped to the same axes after the reflection on the cavity mirrors. The combination of this mapping effect with the Fraunhofer diffraction leads to an increasing of the cavity losses along the direction of the wire. For this reason, when the wire crosses the optical axis we observe a laser pattern with two intensity lobes perpendicular to the wire. If we move the wire away from the center, the inverse symmetry by the axis of the wire is broken. In this case the laser field along the wire is not exactly mapped to the same axis. Therefore, the losses along the wire will be
1©
w
w 5 mm
jFig. 1. Two-lobe pattern obtained after the] Insertion of a wire crossing the optical axis. The| output power is 150 mW
5 mm.
Fig. 2. Recovered fundamental mode alter a 200 urn displacement of the wire away from the| (optical axis. The output power is 150 mW.
much smaller respect to the former case. As a consequence, this time the laser pattern does not present regions of zero intensity as shown in Fig.2. This means that it is possible to recover a condition of circular symmetry for the field with a small shift of the wire from the center. Besides the imdamental mode, we also studied the effect of the wire on the q=l family of taguerre-Gauss modes, selected with a diaphragm aperture of 10.0 mm. Starting from the unperturbed TEM§i# mode (the asterisk denotes two degenerates modes combined in space and in phase quadrature forming a circular symmetric mode) and inserting the wire crossing the optical axis, we obtain the TEM§i mode (Fig. 3), with the two lobes aligned perpendicular to the wire.
w
W
5 mm
Fig. 3, Pattern obtained starting from the annular mode, after the insertion of a wire crossing the! optical axis. The output power is 600 mW
5 mm
JFig. 4. Pattern obtained after a 200 um| displacement of the away from the optical axis. The output power remains unchanged.
164
By moving the wire away from the cavity axis of 200 jjm it is possible to partially recover the symmetry of the unperturbed mode. The result is the superposition of a weak •annular pattern with a TEMi© mode with lobes aligned m the same direction of the wire (Fig. 4). A further displacement of 100 pm determines the cancellation of die weak annular contribution and the appearance of the TEMio pattern oriented along the wire direction (Fig. 5). This orientation confirms the non-trivial role played by the wire in the mode selection.
w 5 mm
Fig. 5. Pattern obtained after a further displacement of 100 >*m with respect to Fig. 4. In] this case the output power is 580 mW
Once studied the case of a single wire, we passed to analyze the effects produced by grids composed by more than one wire. We found that it is possible to select and stabilize different kinds of elementaiy cells, as squares and hexagons, which are of great importance in pattern formation. In particular we studied the spatial bifurcations of the hexagonal patterns, obtained using a mask of thin metallic wires (100pm of diameter) aligned along three directions making an angle of 60° with each other8. As in the case of me single wire, the mask is located at 110 mmfrom"the outcoupier mirror. The spatial perturbation corresponding to the hexagonal mask can be approximated with the form5 f(r) = -le**1 f + elk4'r + elk*>'r + c.c.L where £f- (i=l,4,6) are the spatial wavevectors making an angle of 2n/3 rad with each other. In the experiment the magnitude of £,- is 2n/6 mm"1 (6 mm is me separation between the parallel wires). We use the cavity detuning as the control parameter to explore die different patterns stabilized by the hexagonal mask. We assume the zero reference for die cavity detuning is in correspondence with the simple 6-lobe pattern (Fig.6a). The lobe maxima, observed on a plate at a distance of.. 400 mm far from the outcoupier mirror, are located at a distance of r© = 4.8 mm from the center. By increasing die detuning we observe, after an
165
intermediate configuration (Fig.6b), a transition toward a 12-lobe pattern (Fig.6c). This corresponds to a doubling of the azimuthal spatial frequency with respect to the initial pattern. In this case the distance of the maximafrom,the center is 4r©/3. Then, fiirther increasing the detuning, the 12-lobe pattern loses its stability and, after another intermediate pattern (Fig.6d), we obtain a double hexagon (Fig.6e). In this case, the spatial bifurcation occurs on the radial coordinate. In fact, the distances oftiheinner and
(a) J %
(b) J ^ ^ L
(d) # %
(e)
?£
(,)
!
ii
Fig. 6. Experimental sequence of patterns with hexagonal symmetry, obtained by varying the cavity detuning within one free spectral range (FSR). (a) 0% FSR; (b) 28% FSR; (c) 67% FSR; (d) 79% FSR; (e) 82% F8R; (I) 87% FSR.
the outer peaks from the center are 2r©/3 and 4ro/3, respectively. A further increase of Ihe detuning induces a merging of the radial lobes (Fig.6f), and finally, after a one free spectral range the initial 6-lobe pattern is found again. We also studied the effects of a small misalignment of the outcoupler mirror, mamtaining the mask in the same position. For small values of the tilt angle (< V) the hexagonal symmetry is broken, and we obtain pattern with eight or ten lobes. By increasing the tilting, the hexagonal symmetry is partly recovered, but with more complex structure (Fig. 7). An important issue concerns the temporal behavior. By monitoring the local intensity with a fast HgCdTe detector, we found the presence of complicated temporal
166
oscillations in the unperturbed patterns. The insertion of the mask, besides the spatial pattern stabilization, also provides the elimination of the temporal oscillation. From the above results, we can see that the two important parameters controlling the spatial bifurcations of the hexagonal pattern are the cavity detuning and the tilt angle of the outcoupler mkror. In particular, a change in the alignment of the optical cavity can lead to a change of the hexagonal symmetry imposed by the mask.
The diffraction effects produced by the mask can be taken into account by a numerical simulation of the field propagation in the cavity based on the Fox and Li method. Following this approach we are able to reproduce both the six and the twelve lobe patterns observed in the experiment. 3. Acknowledgments Work partly supported by coordinated project CfiNonlinear dynamics in optical systems" of the Italian National Council of Eesearch and by the European Contact FMRXCT960010 ** Nonlinear dynamics and statistical physics of spatially extended systems". p-YW acknowledges the support of the National Natural Science Foundation of China.
4. References 1. E. Ott, C. Grebogi, and J.A. Yorke, Phys. Rev. Lett. §4 (1990) 1196. 2. R. Martin, AJ. Scroggie, G.-L. Oppo, and W.J. Firth, Phys. Rev, Lett. 77 (1996) 4007; A. V. Mamaev and M. Safftnan, Phys. Rev. Lett. §§ (1998) 3499. 3. KJPyragas Phys. Lett. A 170 (1992) 421.
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4. W. Lu, D. Yu and R. G. Harrison, Phys. Rev. Lett. 76 (1996) 3316; W. Lu, D. Yu and R. G. Harrison, Phys. Rev. Lett. 78 (1997) 4375. 5. Peng-Ye Wang, Ping Xie, Jian-Hua Dai, and Hong-Jun Zhang, Phys. Rev. Lett. 80, (1998)4669. 6. W. W. Rigrod, Appl. Phys. Lett. 2 (1963) 51. 7. M. Ciofini, A. Labate, R. Meucci, and Peng-Ye Wang, Opt. Commun. 154 (1998) 307. 8. R. Meucci, A. Labate, M. Ciofini, and Peng-Ye Wang, Quantum Semiclass. Opt. 10 (1998) 803.
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V. Biology I
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ROBUST DETECTION OF DYNAMICAL CHANGE IN SCALP EEG PAUL C. GAILEY, LEE M. HIVELY, and VLADIMIR A. PROTOPOPESCU Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
ABSTRACT We present a robust, model-independent technique for measuring changes in the dynamics underlying nonlinear time-serial data. We define indicators of dynamical change by comparing distribution functions on the attractor via Lrdistance and x2 statistics. We apply the measures to scalp EEG data with the objective of capturing the transition between nonseizure and epileptic brain activity in a timely, accurate, and non-invasive manner. We find a clear superiority of the new metrics in comparison to traditional nonlinear measures as discriminators of dynamical change.
1. Introduction This work focuses on nonlinear analysis of physiological data. Typically, these data arise from a virtual "black box" with little knowledge of the underlying system, its dimensionality, or noise contamination. More often than not, nonlinear analysis requires some assumptions about the underlying dynamics. For example, calculation of Lyapunov exponents or Kolmogorov entropy implicitly assumes that the physiological process can be modeled as a dynamical system. At a more fundamental level, one may ask whether the data arises from a stationary process. Numerous statistical tests for stationarity exist, but such tests usually assume that the dynamics are stationary within the two time windows under comparison. Moreover, complex systems, like the brain, may not be well modeled by stationary dynamics over long times. We describe a model-independent method for measuring change in nonstationary data. The dynamics of reference and test cases are represented as discrete distributions of the density of points in reconstructed phase space during different time windows. Variability is captured by the visitation frequency at various regions of phase space as described by the distribution function. The method quantifies differences in these reconstructed dynamics by comparing the distribution functions. We make no assumption about stationarity, because no dynamical properties are inferred from the reconstructed attractor. The system dynamics may change within the time window, but such variability presents no problem for our technique, which measures dynamical change over a variety of length scales, and over a wide range of time scales. Moreover, our method allows measurement of dynamical change that that occurs continuously or intermittently. Recently, Moeckel and Murray discussed similar concepts for measuring the "distance" between attractors from time-delay reconstructions. In this context, our method provides continuous measures of change in contrast to stationarity tests for whether or not any statistically significant change has occurred. Due to their continuous nature and their independence from assumptions about stationarity, our measures are particularly useful for analysis of physiological data.
171
172 2. Traditional Nonlinear Measures We assume that an unknown scalar signal, x, is sampled at equal time intervals, t, starting at time, to, yielding a sequence of N points, Xj = x(to + ix). Dynamical process reconstruction2 uses d-dimensional time-delay vectors, y(i)=[xi, XJ+X , ..., Xj+^.i^ ], for a system with d active variables and time lag, X. The choice of lag and embedding dimension, d, determine how well the reconstruction unfolds the dynamics for a finite amount of noisy data. A proper reconstruction allows calculation of nonlinear measures that are consistent with the original dynamics. Below, we use three traditional measures, for comparison to our phase-space indicators of dissimilarity. The mutual information function is a nonlinear form of auto-correlation function. Mutual information was devised by Shannon and Weaver3, and applied to time series by Fraser and Swinney4. Mutual information measures the information (in bits) that can be inferred from one signal about a second signal, and is a function of the time delay between the measurements. Univariate (bivariate) mutual information measures information within the same (different) data stream(s) at different times. Here, we use the first minimum, Mi, in the univariate mutual information function. Mi measures the average time separation (in timesteps) that decorrelates two points in the time series. The correlation dimension measures process complexity and is a function of scale length, 5, in the data. Our choice of length scale balances local dynamics (typically at 8 < 3a) against avoidance of excessive noise (typically at 8 > a). The symbol, a, denotes the absolute average deviation as a robust indicator of variability in the time serial data: N
a = (l/N)Z|xi-x|.
(1)
i=l
The symbol x denotes the mean of x;. We use the maximum-likelihood correlation dimension, D, developed by Takens6 with modifications for noise by Schouten et al.5 The Kolmogorov entropy, K, measures the rate of information loss (bits/s). Positive, finite entropy generally is considered to clearly indicate chaotic features. Large entropy implies a stochastic, totally unpredictable process. Entropy measures the average time for two points on an attractor to evolve from a small initial separation to more than a specific (large) distance, 8 > 80. We use maximum-likelihood entropy by Schouten et al7. Noise corrupts all real data. Also, finite precision computer arithmetic truncates model data. Thus, we choose a finite-scale length that is larger than the noise, So = 2a, at which to report K and D, corresponding to finite-scale dynamical structure. Thus, our values af K and D have smaller values than expected for the zero-scale-length limit. 3. New Measures of Dynamical Change Traditional nonlinear measures characterize global features by averaging or integrating over the data. Such measures describe the long-term behavior but poorly indicate dynamical change. Greater discrimination is possible by more detailed analysis of the reconstructed dynamics. The natural (or invariant) measure on the attractor
173 provides a more refined representation of the reconstruction, describing the visitation frequency of the system dynamics over the phase space. We converted each signal value, Xj, to one of S different integers, {0, 1, ..., S-1}: 0 < Sj = INT[S(Xi - xmin)/(xmax - xmin)] < S-1.
(2)
Here, xmjn and xmax denote the minimum and maximum values of Xj, respectively, over both the reference case and over the test cases. INT is a function that converts a decimal number to the next lower integer. For xmin ^ x, < xmsx, the inequality 0 < Sj < S-l holds trivially. We took Sj(x, = xmax)= S-l in order to maintain exactly S distinct symbols and to partition the phase space into Sd hypercubes or bins. We then discretized the distribution function on the attractor, by counting the number of phase-space points occurring in each bin. We denoted the population of the i-th bin of the distribution function, Pi, for the base case, and Qi for a test case, respectively. For this initial work, we iteratively varied each parameter (S, d, N, etc.) with the others fixed, to obtain optimum sensitivity of the measures to changes in EEG dynamics. A systematic method to determine optimal values for these parameters is the subject of future work. We used an embedding window, Mi = (d - 1)^. Here, the first minimum in the mutual information function, Mi, is measured in timesteps. We obtained an integer value for the reconstruction lag by setting X = INT[0.5 + Mi/(d-l)] > 1, thus constraining the largest value of dimensionality to d < 2Mi + 1. We compared the distribution function of a test state to the reference state, by measuring the difference between Pi with Qi via the %2 statistics and Li distance: X2 =2(Pi-Qi) 2 /(Pi + Qi),and
(3)
i
L = S|P,-Qi|.
(4)
i
The summations include all of the populated cells in the phase space. The sum in the denominator of Eq. 3 is based on a test for equality of two multinomial distributions. Proper application of these measures requires a rescaling so that the total population of the test case distribution function is the same as the total population of the base case. By connecting successive phase-space points as indicated by the dynamics, y(i) —> y(i+l), we constructed a 2d-dimensional phase-space vector, Y(i)=[y(i), y(i+l)]. Thus, we obtained a discrete representation of the process flow.8 This approach extends the method to capture more dynamical information using pair-wise connectivity between successive d-dimensional states. We use base S arithmetic to assign an identifier j = Ii for the i-th phase-space state, using I, = 2dm" Si(m). The sum runs from m=l to m=d, corresponding to successive components of the d-dimensional phase-space vector. The symbol, Sj(m), denotes the mth component of the i-th phase-space vector. The numeric identifier for the sequel phasespace point is k = Ij+i. Then, we can define the measure of the dissimilarity between these two connected phase-space states, as before, via the Li-distance and x 2 statistics:
174
Xc2 = 2(P jk -Qjk) 2 /(Pjk + Qjk),and
(5)
Lc = S|Pj k -Q, k |.
(6)
Pjk and Qjk denote the distribution functions for the basecase and testcase, respectively, in the connected phase space. The summations in both equations run over all of the populated cells in the connected phase space. The subscript, c, denotes the connected measures, which are stronger metrics than the non-connected versions, according to the following inequalities9"10: %2 < L, %2 < Lc, L < Lc, and %2 < %2. We tested the discriminating power of our measures on chaotic regimes of the Lorenz system9 and of the Bondarenko model.10"11 The latter model mimics highdimensional EEG dynamics via a system of delay-differential equations. Over a broad parameter range, the phase-space measures increased monotonically by more than four orders of magnitude. Over this same range, traditional nonlinear measures were indistinguishable from noise or varied erratically by a factor of two. These results gave us confidence that the phase-space measures would be useful for noisy clinical EEG data. 4. EEG Analysis and Results We converted one channel of analog EEG data on VHS tapes to 12-bit digital form at a sampling rate of 512 Hz. We chose N=20480 data points for each cutset. This choice balances better time discrimination (smaller N) against higher statistical power (larger N). We used the first 400 seconds of data to construct ten non-overlapping 40second basecase cutsets. We compared each base case cutset to every test case cutset to obtain average values for % and L (and a corresponding standard deviation of the mean). We overlapped adjacent test case cutsets by 50% for smooth time-history trending. We also removed muscular artifacts (e.g., eye blinks) with a zero-phase quadratic filter.9"10 We found that d=3 and S=34 were adequate for our EEG data. The value of Mi came from the first 400 seconds of (nonseizure) data. However, the disparate range and variability of the conventional and phase-space measures were difficult to interpret. Thus, we renormalized the nonlinear measures. For each nonlinear measure, V, we defined Vj as the value of nonlinear measure for the i-th cutset. The variable, V, was in turn D, K, Mi, %2, etc. We obtained the mean, V, of Vj over ten non-overlapping cutsets (each with N=20480) for the first 400 seconds (base case interval) of the dataset. The corresponding sample standard deviation was denoted by o\ Then, the renormalized form was U(V) = |Vi - V|/o\ For an indication of change, we used U > U c = 4.265, corresponding to a false positive probability of )2, vA = 2dxadx()) + adxx(j), v5 — a\a, and set II, corresponding to Eq. (11): v0 = adt, V\ = adx, v2 — a, v3 = 2dxadxcp + adxxct>, v 4 = dxxa — a(dx(f))2, and v5 = a[a. Numerical studies on several dynamical model equations [11] revealed that the CGLE could be estimated with high accuracy from noise-free data, leading to a maximal correlation of almost unity. As a first question, we want to check whether the spatiotemporal evolution of the system can be described by the coupled CGLEs (10,11). In this case, one expects the following optimal transformations: The function $ 0 should be the identity, $ 2 should be a third-order polynomial in a, and all the other functions should be linear, with slopes corresponding to the coefficients in Eqs. (10,11). As a check, below our results will be compared with experimentally obtained coefficients [12]. In [12] it was also shown that most of the experimental values agree reasonably well with the ones calculated from first principles. These experimental values are represented as smooth curves in Fig. 4. Since the polynomials ETQ1 a + gr^1 a3 and SCQTQ1 a + gc-iT^1 a3 have large uncertainties, the curves representing their extremal values are shown in the upper and lower panels for $2, respectively. The distribution of the amplitudes, phases, and derivatives are rather inhomogeneous with heavy tails. Therefore, in Fig. 4 the range on the abscissa that is covered by 96% of the data values is marked by vertical dotted lines. Since the optimal transformations are harder to estimate for very sparse data, each 2% of the transformed data values at the edges are considered as outliers. For the seven analyzed data sets we obtain the following results (Fig. 4): For large bifurcation parameters (e' > 12.07), the expected functions coincide quite well with the coefficients found in [12]. In particular: Set I (top row of Fig. 4): The estimate for the left-hand side, o> turns out to be approximately the identity; the estimate for $1 is an approximately linear function in dxa with a slope in good agreement with the wave velocity s measured in [12]; the estimate for $ 2 can be described by a cubic polynomial in a; the estimates for $3 and $ 4 are approximately linear, also with correct slopes. The estimate for the coupling term, $5, appears to be approximately
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Figure 4: Estimated optimal transformations for the set of terms I and II, both for e' = 12.07 (upper two rows) and for e' — 1.77 (lower two rows). The ordinates are the optimal transformations multiplied by 1000. They are the same for all plots in one row, except in the frames for $ 5 where they have been magnified by 2.5. The abscissae are given by the terms v0 to v5, respectively, and are not labeled for clarity. Additionally, smooth curves indicate the theoretically expected functions, and vertical dotted lines mark the ranges on the abscissae where 96% of the data values are located, as explained in the text. The results for e' = 14.03 and 16.28 resemble the results for e' = 12.07 and are therefore not shown; similarly, the results for e' = 4.22, 6.38, and 9.32 resemble the results for e' = 1.77. (From [14].)
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linear in a[a with clearly negative coupling coefficient. Set II (second row of Fig. 4) yields similar results, but obviously the estimates for <J>3 and BCL, and thus the element (25,25) is still in a refractory state (state 2) when the second stimulus is given. By the time the third stimulus is delivered, however, the tissue has returned to its excitable state (state 0), and is now ready to respond, as shown in (e,f). The patterns shown in Figs. 1 and 2 are bistable — they coexist for the same set of parameter values, and the system falls into one pattern or another depending on initial conditions.
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Fig. 1 Snapshots of a simulation with <Ti=0.60. All elements (ij) are initially set with APDij°=118.5 msec, the steady state value on the N=l branch at BCL=150 msec. Periodic stimulation at BCL=150 msec is delivered from element (25,25). In Frame la, a small region of elements centered about (25,25) has become excited (state 1, black region), while the rest of the tissue is unexcited (state 0, white region). In frame lb, at t=90 msec, the excited region has spread to the boundaries of the "tissue", and the center region is now in a refractory state (state 2, grey region). When another stimulus is delivered at t=150 msec, the tissue is ready to respond (frames c and d).
Reentrant Waves, Phase Singularities and Rotors: A Small Region of 2:1 Suppose that a small region of the map, though in phase with its neighbors, has fallen onto the initial conditions for another branch. Specifically, suppose we have the entire tissue initially in a 1:1 state, except for a small region in a 2:1 state. In order for the 2:1 region to remain active until the surrounding 1:1 wavefront has passed, we must require Oi(APD2+0)> APD,+e. If this condition is met, the active 2:1 elements will "reenter" the recovered region in the wake of the passing front, and restimulate the just-recovered 1:1 elements.
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Fig. 2 Snapshots from a simulation with conditions identical to Fig. 1, except that the initial conditions APD jj° are set to 182 msec, the steady state value of APD on the N=2 branch at BCL=150 msec. Since APD+9 >BCL, the tissue at the pacing site (25,25) is still refrartory at t=150 msec, as shown in Fig 2c. No response is elicited by a stimulus delivered at this point (marked with an x). By t=240, the tissue has recovered, shown in Fig. 2d. At t=2BCL=300 msec, the tissue is ready to respond again to an apphed stimulus. This illustrates a 2:1 pattern, in contrast to the 1:1 pattern shown in Fig 1.
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Fig. 3. Phase resetting plots for a simulation in which the "1:1" elements are set with APDij°=78.85 msec, a value in the basin of attraction of the 1:1 fixed point (APD=103.82 msec) at BCL=120 msec. A strip of elements (j=75 to 78, i=20 to 80) is set with APDij°=158.63 msec, the 2:1 fixed point at this BCL. Periodic stimulation is delivered from element [25][25] at BCL=120 msec. Panels show the evolution of P;/ and APDi/ at (a) the stimulation site, (b) an element in the reentrant 2:1 region and (c) and an element initially on the "1:1" (or "transient 2:2") branch. By t=200 msec, all elements are being periodically stimulated by the rotor at intervals Tf. Coordinates [i][j] are shown in the top right corner of each panel.
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Fig. 4 A phase singularity in the simulation illustrated in Fig. 3, at t=235 msec. Because of the size and location of the strip of 2:1 elements with respect to the stimulation site, reexcitation of the recovered tissue by the 2:1 elements generates a clearly visible pair of "twin rotors". The locations of the rotor cores do not appear to drift during the duration of the simulation. Color code: black: 0